Deformations of Linear Lie Brackets

TL;DR

This paper introduces a differential graded Lie algebra associated with VB-algebroids that governs their deformations, providing new insights into the structure and applications of these geometric objects.

Contribution

It constructs a differential graded Lie algebra for VB-algebroids and demonstrates its role in controlling their deformations, advancing the theory of Lie algebroid structures.

Findings

The differential graded Lie algebra controls VB-algebroid deformations.

Examples illustrate the application of the theory.

Foundation for future work on deformations over differentiable stacks.

Abstract

A VB-algebroid is a vector bundle object in the category of Lie algebroids. We attach to every VB-algebroid a differential graded Lie algebra and we show that it controls deformations of the VB-algebroid structure. Several examples and applications are discussed. This is the first in a series of papers devoted to deformations of vector bundles and related structures over differentiable stacks.